the new fuels with magnecular structure - Institute for Basic Research

THE NEW FUELS WITH MAGNECULAR STRUCTURE 175
In the atomic units (e = = m = 1), using the notation
E ′ = 2 γ
m + E, n2 = 1
−2E ′ ,
(A.20)
introducing the new variable x = 2z/n, and dropping x 0 = 2z 0 /n, to simplify
representation, the above equation can be rewritten as
[ d
(−
dx 2 + 1 4 + n )]
χ(x) = 0,
(A.21)
x
where x > 0 is assumed. Introducing new function v(x) defined as χ(x) =
xe −x/2 v(x), one gets the final form of the equation,
xv ′′ + (2 − x)v ′ − (1 − n)v = 0.
(A.22)
Noting that it is a particular case of Cummer’s equation,
xv ′′ + (b − x)v ′ − av = 0,
(A.23)
the general solution is given by
v(x) = C 1 1 F 1 (a, b, x) + C 2 U(a, b, x),
(A.24)
where
and
1F 1 (a, b, x) =
Γ(b)
Γ(b − a)Γ(a)
U(a, b, x) = 1
Γ(a)
∫ ∞
0
∫ 1
0
e xt t a−1 (1 − t) b−a−1 dt
e −xt t a−1 (1 + t) b−a−1 dt
(A.25)
(A.26)
are the confluent hypergeometric functions, and C 1,2 are constants; a = 1 − n
and b = 2. Hence, for χ(x) one has
χ(x) = (|x|+x 0 )e −(|x|+x 0)/2 [ C ± 1 1F 1 (1 − n, 2, |x| + x 0 ) + C ± 2 U(1 − n, 2, |x| + x 0) ] ,
(A.27)
where the parameter x 0 has been restored, and the “±” sign in C ± 1,2 corresponds
to the positive and negative values of x, respectively (the modulus sign is used
for brevity).
Let us consider first the x 0 = 0 case. The first hypergeometric function 1 F 1 (1−
n, 2, x) is finite at x = 0 for any n. At big x, it diverges exponentially, unless
n is an integer number, n = 1, 2, . . . , at which case it diverges polynomially.